A witnessed multiplicative inverse cancels on the constructed complex numbers

Identifier: F:complex-inv-mul-cancel
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For a constructed complex number x0 with a positivity witness x2 that its squared norm is bounded away from 0 at precision x1, the inverse Complex.inv(x0, x1, x2) times x0 is Complex.Equiv-equivalent to 1.

Formal statement

theorem Complex.inv_mul_cancel : ((x0 : Complex) -> ((x1 : AxNat) -> ((x2 : CReal.PosBound (Complex.normSq x0) x1) -> Complex.Equiv (Complex.mul (Complex.inv x0 x1 x2) x0) Complex.one)))

Dependencies

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3 direct dependencies 2 direct dependents

Evidence

kernel-Complex.inv_mul_cancel

Kind: kernel-term
Status: checked

Supports: Complex.inv_mul_cancel is admitted by the trusted kernel gate with the type recorded in formal.statement.

Checker command

cargo run -q --release -p axeyum-lean-kernel --example theorem_dependency_inventory -- Complex.inv_mul_cancel 2>/dev/null | grep -cE '^Complex\.inv_mul_cancel[[:space:]]'

Evidence notes

`build_complex_prelude` admits Complex.inv_mul_cancel through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type, so a line in this tool's output for the exact name is a machine-checked proof having been admitted. `theorem_dependency_inventory` builds creal/complex/cpoint in one run; it exits non-zero for a named filter that matches nothing, and `grep -c` (never `-q`) independently asserts the exact tab-anchored line is present. `--release` is MANDATORY: building the constructed carriers recurses deep enough in a debug build to overflow the default thread stack (measured on this tree: release exits 0, debug SIGABRTs at 134).

footprint-Complex.inv_mul_cancel

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the complex prelude's trusted surface is empty, which bounds Complex.inv_mul_cancel

Checker command

cargo run -q --release -p axeyum-lean-kernel --example nat_axiom_inventory -- --include-constructed --require-axiom-free complex

Evidence notes

`nat_axiom_inventory --include-constructed --require-axiom-free complex` builds the complex environment as its own group and reports `complex: axiom=0 opaque=0 quotient=0 total_trusted=0` (re-measured on this tree), covering Axiom, Opaque AND Quotient, not just Declaration::Axiom -- this bounds every individual complex theorem's footprint by []. `--include-constructed` is REQUIRED: without it the command exits 1 naming `complex` as known but not built this run rather than a silent pass, which is the standing trap of an empty/absent result reading as a strong negative when the tool was never pointed at the subject. `nat_axiom_inventory` now covers `cpoint` directly (re-verified on this tree: `--require-axiom-free cpoint` exits 0 with `cpoint: axiom=0 opaque=0 quotient=0 total_trusted=0`), so the footprint_closure_audit fallback some earlier CPoint facts used is not needed here. `--release` is MANDATORY for the same deep-recursion reason as `theorem_dependency_inventory` (measured on this tree: debug SIGABRTs at 134).

Provenance

{
  "date": "2026-08-25",
  "established_by": "axeyum-lean-kernel build_complex_prelude",
  "source": "theorem name and dependency edges from theorem_dependency_inventory (builds creal/complex/cpoint); canonical type and top-level binder arity read via a standalone probe binary depending on axeyum-lean-kernel by path, calling only its public Kernel API (environment(), display_name(), render_lean(), axiom_footprint(), expr_node()) -- no in-tree example prints Complex/CPoint theorem types; crates/ source was not touched to produce this batch; the probe was built and run in the session scratchpad and deleted after use."
}